Integrand size = 23, antiderivative size = 138 \[ \int \frac {A+B \sec (c+d x)}{(a+a \sec (c+d x))^4} \, dx=\frac {A x}{a^4}-\frac {(55 A-6 B) \tan (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac {2 (80 A-3 B) \tan (c+d x)}{105 a^4 d (1+\sec (c+d x))}-\frac {(A-B) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac {(10 A-3 B) \tan (c+d x)}{35 a d (a+a \sec (c+d x))^3} \]
[Out]
Time = 0.31 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {4007, 4004, 3879} \[ \int \frac {A+B \sec (c+d x)}{(a+a \sec (c+d x))^4} \, dx=-\frac {2 (80 A-3 B) \tan (c+d x)}{105 a^4 d (\sec (c+d x)+1)}-\frac {(55 A-6 B) \tan (c+d x)}{105 a^4 d (\sec (c+d x)+1)^2}+\frac {A x}{a^4}-\frac {(10 A-3 B) \tan (c+d x)}{35 a d (a \sec (c+d x)+a)^3}-\frac {(A-B) \tan (c+d x)}{7 d (a \sec (c+d x)+a)^4} \]
[In]
[Out]
Rule 3879
Rule 4004
Rule 4007
Rubi steps \begin{align*} \text {integral}& = -\frac {(A-B) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac {\int \frac {-7 a A+3 a (A-B) \sec (c+d x)}{(a+a \sec (c+d x))^3} \, dx}{7 a^2} \\ & = -\frac {(A-B) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac {(10 A-3 B) \tan (c+d x)}{35 a d (a+a \sec (c+d x))^3}+\frac {\int \frac {35 a^2 A-2 a^2 (10 A-3 B) \sec (c+d x)}{(a+a \sec (c+d x))^2} \, dx}{35 a^4} \\ & = -\frac {(55 A-6 B) \tan (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac {(A-B) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac {(10 A-3 B) \tan (c+d x)}{35 a d (a+a \sec (c+d x))^3}-\frac {\int \frac {-105 a^3 A+a^3 (55 A-6 B) \sec (c+d x)}{a+a \sec (c+d x)} \, dx}{105 a^6} \\ & = \frac {A x}{a^4}-\frac {(55 A-6 B) \tan (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac {(A-B) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac {(10 A-3 B) \tan (c+d x)}{35 a d (a+a \sec (c+d x))^3}-\frac {(2 (80 A-3 B)) \int \frac {\sec (c+d x)}{a+a \sec (c+d x)} \, dx}{105 a^3} \\ & = \frac {A x}{a^4}-\frac {(55 A-6 B) \tan (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac {(A-B) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac {(10 A-3 B) \tan (c+d x)}{35 a d (a+a \sec (c+d x))^3}-\frac {2 (80 A-3 B) \tan (c+d x)}{105 d \left (a^4+a^4 \sec (c+d x)\right )} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(329\) vs. \(2(138)=276\).
Time = 4.28 (sec) , antiderivative size = 329, normalized size of antiderivative = 2.38 \[ \int \frac {A+B \sec (c+d x)}{(a+a \sec (c+d x))^4} \, dx=\frac {\sec \left (\frac {c}{2}\right ) \sec ^7\left (\frac {1}{2} (c+d x)\right ) \left (3675 A d x \cos \left (\frac {d x}{2}\right )+3675 A d x \cos \left (c+\frac {d x}{2}\right )+2205 A d x \cos \left (c+\frac {3 d x}{2}\right )+2205 A d x \cos \left (2 c+\frac {3 d x}{2}\right )+735 A d x \cos \left (2 c+\frac {5 d x}{2}\right )+735 A d x \cos \left (3 c+\frac {5 d x}{2}\right )+105 A d x \cos \left (3 c+\frac {7 d x}{2}\right )+105 A d x \cos \left (4 c+\frac {7 d x}{2}\right )-9940 A \sin \left (\frac {d x}{2}\right )+1260 B \sin \left (\frac {d x}{2}\right )+8260 A \sin \left (c+\frac {d x}{2}\right )-1260 B \sin \left (c+\frac {d x}{2}\right )-7140 A \sin \left (c+\frac {3 d x}{2}\right )+882 B \sin \left (c+\frac {3 d x}{2}\right )+3780 A \sin \left (2 c+\frac {3 d x}{2}\right )-630 B \sin \left (2 c+\frac {3 d x}{2}\right )-2800 A \sin \left (2 c+\frac {5 d x}{2}\right )+294 B \sin \left (2 c+\frac {5 d x}{2}\right )+840 A \sin \left (3 c+\frac {5 d x}{2}\right )-210 B \sin \left (3 c+\frac {5 d x}{2}\right )-520 A \sin \left (3 c+\frac {7 d x}{2}\right )+72 B \sin \left (3 c+\frac {7 d x}{2}\right )\right )}{13440 a^4 d} \]
[In]
[Out]
Time = 0.74 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.64
method | result | size |
parallelrisch | \(\frac {\left (15 A -15 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}+\left (-105 A +63 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+\left (385 A -105 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (-1575 A +105 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+840 A x d}{840 a^{4} d}\) | \(89\) |
norman | \(\frac {\frac {A x}{a}+\frac {\left (A -B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{56 a d}-\frac {\left (5 A -3 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{40 a d}+\frac {\left (11 A -3 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{24 a d}-\frac {\left (15 A -B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a d}}{a^{3}}\) | \(112\) |
derivativedivides | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} A}{7}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} B}{7}-\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} A +\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} B}{5}+\frac {11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} A}{3}-\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} B -15 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) A +\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) B +16 A \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d \,a^{4}}\) | \(130\) |
default | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} A}{7}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} B}{7}-\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} A +\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} B}{5}+\frac {11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} A}{3}-\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} B -15 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) A +\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) B +16 A \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d \,a^{4}}\) | \(130\) |
risch | \(\frac {A x}{a^{4}}-\frac {2 i \left (420 A \,{\mathrm e}^{6 i \left (d x +c \right )}-105 B \,{\mathrm e}^{6 i \left (d x +c \right )}+1890 A \,{\mathrm e}^{5 i \left (d x +c \right )}-315 B \,{\mathrm e}^{5 i \left (d x +c \right )}+4130 A \,{\mathrm e}^{4 i \left (d x +c \right )}-630 B \,{\mathrm e}^{4 i \left (d x +c \right )}+4970 A \,{\mathrm e}^{3 i \left (d x +c \right )}-630 B \,{\mathrm e}^{3 i \left (d x +c \right )}+3570 A \,{\mathrm e}^{2 i \left (d x +c \right )}-441 B \,{\mathrm e}^{2 i \left (d x +c \right )}+1400 \,{\mathrm e}^{i \left (d x +c \right )} A -147 B \,{\mathrm e}^{i \left (d x +c \right )}+260 A -36 B \right )}{105 d \,a^{4} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{7}}\) | \(181\) |
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.31 \[ \int \frac {A+B \sec (c+d x)}{(a+a \sec (c+d x))^4} \, dx=\frac {105 \, A d x \cos \left (d x + c\right )^{4} + 420 \, A d x \cos \left (d x + c\right )^{3} + 630 \, A d x \cos \left (d x + c\right )^{2} + 420 \, A d x \cos \left (d x + c\right ) + 105 \, A d x - {\left (4 \, {\left (65 \, A - 9 \, B\right )} \cos \left (d x + c\right )^{3} + {\left (620 \, A - 39 \, B\right )} \cos \left (d x + c\right )^{2} + {\left (535 \, A - 24 \, B\right )} \cos \left (d x + c\right ) + 160 \, A - 6 \, B\right )} \sin \left (d x + c\right )}{105 \, {\left (a^{4} d \cos \left (d x + c\right )^{4} + 4 \, a^{4} d \cos \left (d x + c\right )^{3} + 6 \, a^{4} d \cos \left (d x + c\right )^{2} + 4 \, a^{4} d \cos \left (d x + c\right ) + a^{4} d\right )}} \]
[In]
[Out]
\[ \int \frac {A+B \sec (c+d x)}{(a+a \sec (c+d x))^4} \, dx=\frac {\int \frac {A}{\sec ^{4}{\left (c + d x \right )} + 4 \sec ^{3}{\left (c + d x \right )} + 6 \sec ^{2}{\left (c + d x \right )} + 4 \sec {\left (c + d x \right )} + 1}\, dx + \int \frac {B \sec {\left (c + d x \right )}}{\sec ^{4}{\left (c + d x \right )} + 4 \sec ^{3}{\left (c + d x \right )} + 6 \sec ^{2}{\left (c + d x \right )} + 4 \sec {\left (c + d x \right )} + 1}\, dx}{a^{4}} \]
[In]
[Out]
none
Time = 0.32 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.46 \[ \int \frac {A+B \sec (c+d x)}{(a+a \sec (c+d x))^4} \, dx=-\frac {5 \, A {\left (\frac {\frac {315 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {77 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {21 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {3 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{4}} - \frac {336 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{4}}\right )} - \frac {3 \, B {\left (\frac {35 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {35 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {21 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {5 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}\right )}}{a^{4}}}{840 \, d} \]
[In]
[Out]
none
Time = 0.33 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.12 \[ \int \frac {A+B \sec (c+d x)}{(a+a \sec (c+d x))^4} \, dx=\frac {\frac {840 \, {\left (d x + c\right )} A}{a^{4}} + \frac {15 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 15 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 105 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 63 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 385 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 105 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 1575 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 105 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{28}}}{840 \, d} \]
[In]
[Out]
Time = 14.02 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.18 \[ \int \frac {A+B \sec (c+d x)}{(a+a \sec (c+d x))^4} \, dx=\frac {A\,x}{a^4}-\frac {\left (\frac {52\,A\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{21}-\frac {12\,B\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{35}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+\left (\frac {23\,B\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{70}-\frac {16\,A\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{21}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\left (\frac {5\,A\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{28}-\frac {9\,B\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{70}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-\frac {A\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{56}+\frac {B\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{56}}{a^4\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7} \]
[In]
[Out]